Therefore, [itex]\tau[/itex] is the proper time for a futute-directed timelike worldline.

Note also that when [itex]\tau = -1[/itex], both [itex]t[/itex] and [itex]x[/itex] are finite, but as [itex]\tau \rightarrow 0_-[/itex], both [itex]t[/itex] and [itex]x[/itex] wander off to positive infinity.

The situation is unphysical because the 4-acceleration is unbounded, although there are no hyperlight speeds.

Regards,
George

PS I think I have found an expression for the 4-acceleration of a specific example of pervect's problem, but I have to check to see if my solution really does satisfy the necessary criteria.